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Beginning at point $A$ in the diagram below, Dora selects one of the four possible directions with equal probability. Each time she comes to an intersection, she again randomly selects one of the possible directions. What is the probability that, in her first four steps, she will walk completely around the gray square? Express your answer as a common fraction.

Guest Apr 15, 2015

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 #1
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On her first step she has a probability of 1/2 of choosing a direction alongside the gray square.  For each of three steps thereafter she has a probability of 1/4 of choosing a direction that keeps her beside the gray square; so her probability is (1/2)*(1/4)*(1/4)*(1/4) = 1/128

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Alan  Apr 16, 2015
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 #1
avatar+26329 
+10
Best Answer

On her first step she has a probability of 1/2 of choosing a direction alongside the gray square.  For each of three steps thereafter she has a probability of 1/4 of choosing a direction that keeps her beside the gray square; so her probability is (1/2)*(1/4)*(1/4)*(1/4) = 1/128

.

Alan  Apr 16, 2015

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